When you download an e-book through Amazon for your Kindle, or via a service such as Smashwords.com (featuring formats for Nook, Sony eReader, Palm devices, and more), when you buy the book, it's yours. A good example of this is Smashwords, which allows you to download your purchased eBook in multiple formats, whether for your eReader of your PC. And, should something happen to your download, you can download it again from the site.
Smashwords does recommend that instead of sharing an eBook, that you encourage another person to purchase their own copy. Amazon Kindle does allow sharing.
I have written and published eBooks through both services, and am quite pleased with them.
New to ebooks. If I buy an ebook, do I own it or am I just renting it for a period of time? Can I download to a computer and transfer it to someone else? If I purchase a new ebook reader can I transfer existing books to the new reader?